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Topic: Leaving 0^0 undefined -- A number-theoretic rationale
Replies: 48   Last Post: Sep 15, 2013 1:06 PM

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 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: Leaving 0^0 undefined -- A number-theoretic rationale
Posted: Sep 12, 2013 9:15 AM

On Thursday, September 12, 2013 3:46:49 AM UTC-4, Peter Percival wrote:
> Dan Christensen wrote:
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> > On Wednesday, September 11, 2013 5:09:52 PM UTC-4, Peter Percival wrote:
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> >> Dan Christensen wrote:
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> >>>
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> >>> Show me a contradiction that arises from 0^0 = 0.
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> >> The product of the empty set is 1, hence 0^0 = 1. That contradicts 0^0
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> >> = 0. Therefore 0^0 doesn't = 0.
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> > More convincing would be obtaining a contraction by assuming only 0^0=0 along with the usual rules of natural-number arithmetic, including the Laws of Exponents. THAT would be interesting.
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> Among the usual rules is 0^0=1.
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I don't think so. In any case, as I have suggested, it can be shown, without making that assumption, that there are only 2 binary functions on the natural numbers that satisfy the Laws of Exponents -- one has 0^0=1, the other 0^0=0. They give identical results at every other point.

Dan